1
Problems
The following are some problems that will help you refresh your
memory about material you should have learned in undergraduate
chemistry classes and that allow you to exercise the material taught in
this text.
Suggestions about what you should be able to do relative to the background
material in the Chapters of Part 1
1. You should be able to set up and solve the one- and two-dimensional particle in a box
Schr?dinger equations. I suggest you now try this and make sure you see:
a. How the second order differential equations have two independent solutions, so the
most general solution is a sum of these two.
b. How the two boundary conditions reduce the number of acceptable solutions from
two to one and limit the values of E that can be “allowed”.
c. How the wave function is continuous even at the box boundaries, but dΨ/dx is not. In
general dΨ/dx, which relates to the momentum because – i h d/dx is the momentum
operator, is continuous except at points where the potential V(x) undergoes an infinite
jump as it does at the box boundaries. The infinite jump in V, when viewed
classically, means that the particle would undergo an instantaneous reversal in
2
momentum at this point, so its momentum would not be continuous. Of course, in any
realistic system, V does not have infinite jumps, so momentum will vary smoothly
and thus dΨ/dx will be continuous.
d. How the energy levels grow with quantum number n as n2.
e. What the wave functions look like when plotted.
2. You should go through the various wave functions treated in the Part 1 (e.g., particles
in boxes, rigid rotor, harmonic oscillator) and make sure you see how the |Ψ|2 probability
plots of such functions are not at all like the classical probability distributions except
when the quantum number is very large.
3. You should make sure you understand how the time evolution of an eigenstate Ψ
produces a simple exp(-i tE/ h) multiple of Ψ so that |Ψ|2 does not depend on time.
However, when Ψ is not an eigenstate (e.g., when it is a combination of such states), its
time propagation produces a Ψ whose |Ψ|2 probability distribution changes with time.
4. You should notice that the densities of states appropriate to the 1-, 2-, and 3-
dimensional particle in a box problem (which relate to translations in these dimensions)
depend of different powers of E for the different dimensions.
5. You should be able to solve 2x2 and 3x3 Hückel matrix eigenvalue problems both to
obtain the orbital energies and the normalized eigenvectors. For practice, try to do so for
3
a. the allyl radical’s three pi orbitals
b. the cyclopropenly radical’s three pi orbitals.
Do you see that the algebra needed to find the above sets of orbitals is exactly the same as
was needed when we treat the linear and triangular sodium trimer?
6. You should be able to follow the derivation of the tunneling probability. Doing this
offers a good test of your ability to apply the boundary conditions properly, so I suggest
you do this task. You should appreciate how the tunneling probability decays
exponentially with the “thickness” of the tunneling barrier and with the “height” of this
4
barrier and that tunneling for heavier particles is less likely than for light particles. This is
why tunneling usually is considered only for electrons, protons, and neutrons.
7. I do not expect that you could carry off a full solution to the Schr?dinger equation for
the hydrogenic atom. However, I think you need to pay attention to
a. How separations of variables leads to a radial and two angular second order
differential equations.
b. How the boundary condition that φ and φ + 2pi are equivalent points in space
produces the m quantum number.
c. How the l quantum number arises from the θ equation.
d. How the condition that the radial wave function not “explode” (i.e., go to infinity) as
the coordinate r becomes large gives rise to the equation for the energy E.
e. The fact that the angular parts of the wave functions are spherical harmonics, and that
these are exactly the same wave functions for the rotational motion of a linear
molecule.
f. How the energy E depends on the n quantum number as n-2 and on the nuclear charge
Z as Z2, and that the bound state energies are negative (do you understand what this
means? That is, what is the zero or reference point of energy?).
8. You should make sure that you are familiar with how the rigid-rotor and harmonic
oscillator energies vary with quantum numbers (J, M in the former case, v in the latter).
You should also know how these energies depend on the molecular geometry (in the
5
former) and on the force constant and reduced mass (in the latter). You should note that
E depends quadratically on J but linearly on v.
9. You should know what the Morse potential is and what its parameters mean. You
should understand that the Morse potential displays anharmonicity but the harmonic
potential does not.
10. You should be able to follow how the mass-weighted Hessian matrix can be used to
approximate the vibrational motions of a polyatomic molecule. And, you should
understand how the eigenvalues of this matrix produce the harmonic vibrational
frequencies and the corresponding eigenvectors describe the motions of the molecule
associated with these frequencies.
6
Practice with matrices and operators.
1. Find the eigenvalues and corresponding normalized eigenvectors of the following
matrices:
??
?
??
?-1 2
2 2
??
??
??
??-2 0 0 0 -1 2
0 2 2
2. Replace the following classical mechanical expressions with their corresponding
quantum mechanical operators:
K.E. = mv
2
2 in three-dimensional space.
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p = mv, a three-dimensional Cartesian vector.
y-component of angular momentum: Ly = zpx - xpz.
3. Transform the following operators into the specified coordinates:
Lx = h
?
i ???
??
??
??
?
y ??z - z ??y from Cartesian to spherical polar coordinates.
Lz = h
-
i
?
?φ from spherical polar to Cartesian coordinates.
4. Match the eigenfunctions in column B to their operators in column A. What is the
eigenvalue for each eigenfunction?
Column A Column B
i. (1-x2) d
2
dx2 - x
d
dx 4x4 - 12x2 + 3
ii. d
2
dx2 5x4
iii. x ddx e3x + e-3x
8
iv. d
2
dx2 - 2x
d
dx x2 - 4x + 2
v. x d
2
dx2 + (1-x)
d
dx 4x3 - 3x
Review of shapes of orbitals
5. Draw qualitative shapes of the (1) s, (3) p and (5) d atomic orbitals (note that these
orbitals represent only the angular portion and do not contain the radial portion of the
hydrogen like atomic wave functions) Indicate with ± the relative signs of the wave
functions and the position(s) (if any) of any nodes.
6. Plot the radial portions of the 4s, 4p, 4d, and 4f hydrogen like atomic wave functions.
7. Plot the radial portions of the 1s, 2s, 2p, 3s, and 3p hydrogen like atomic wave
functions for the Si atom using screening concepts for any inner electrons.
9
Labeling orbitals using point group symmetry
8. Define the symmetry adapted "core" and "valence" atomic orbitals of the following
systems:
NH3 in the C3v point group,
H2O in the C2v point group,
H2O2 (cis) in the C2 point group
N in D∞h, D2h, C2v, and Cs point groups
N2 in D∞h, D2h, C2v, and Cs point groups.
A problem to practice the basic tools of the Schr?dinger equation.
9. A particle of mass m moves in a one-dimensional box of length L, with boundaries at x
= 0 and x = L. Thus, V(x) = 0 for 0 ≤ x ≤ L, and V(x) = ∞ elsewhere. The normalized
10
eigenfunctions of the Hamiltonian for this system are given by Ψn(x) = ??
?
??
?2
L
1/2 Sinnpix
L ,
with En = n
2pi2h?2
2mL2 , where the quantum number n can take on the values n=1,2,3,....
a. Assuming that the particle is in an eigenstate, Ψn(x), calculate the probability that the
particle is found somewhere in the region 0 ≤ x ≤ L4 . Show how this probability
depends on n.
b. For what value of n is there the largest probability of finding the particle in
0 ≤ x ≤ L4 ?
c. Now assume that Ψ is a superposition of two eigenstates,
Ψ = aΨn + bΨm, at time t = 0.
What is Ψ at time t? What energy expectation value does Ψ have at time t and how does
this relate to its value at t = 0?
d. For an experimental measurement which is capable of distinguishing systems in state
Ψn from those in Ψm, what fraction of a large number of systems each described by
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Ψ will be observed to be in Ψn? What energies will these experimental
measurements find and with what probabilities?
e. For those systems originally in Ψ = aΨn + bΨm which were observed to be in Ψn at
time t, what state (Ψn, Ψm, or whatever) will they be found in if a second
experimental measurement is made at a time t' later than t?
f. Suppose by some method (which need not concern us at this time) the system has
been prepared in a nonstationary state (that is, it is not an eigenfunction of H). At the
time of a measurement of the particle's energy, this state is specified by the
normalized wave function Ψ = ??? ???30L5 1/2x(L-x) for 0 ≤ x ≤ L, and Ψ = 0 elsewhere.
What is the probability that a measurement of the energy of the particle will give the
value En = n
2pi2h?2
2mL2 for any given value of n?
g. What is the expectation value of H, i.e. the average energy of the system, for the
wave function Ψ given in part f?
A problem on the properties of non-stationary states
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10. Show that for a system in a non-stationary state,
Ψ = ∑
j
CjΨje-iEjt/h- , the average value of the energy does not vary with time but the
expectation values of other properties do vary with time.
A problem about Jahn-Teller distortion
11. The energy states and wave functions for a particle in a 3-dimensional box whose
lengths are L1, L2, and L3 are given by
E(n1,n2,n3) = h
2
8m ??
?
??
?
??
?
??
?n1
L1
2 +
??
?
??
?n2
L2
2 +
??
?
??
?n3
L3
2 and
Ψ(n1,n2,n3) = ??
?
??
?2
L1
1
2 ??? ???2
L2
1
2 ??? ???2
L3
1
2 Sin
??
?
??
?n1pix
L1 Sin??
?
??
?n2piy
L2 Sin??
?
??
?n3piz
L3 .
These wave functions and energy levels are sometimes used to model the motion of
electrons in a central metal atom (or ion) which is surrounded by six ligands in an
octahedral manner.
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a. Show that the lowest energy level is nondegenerate and the second energy level is
triply degenerate if L1 = L2 = L3. What values of n1, n2, and n3 characterize the states
belonging to the triply degenerate level?
b. For a box of volume V = L1L2L3, show that for three electrons in the box (two in the
nondegenerate lowest "orbital", and one in the next), a lower total energy will result if
the box undergoes a rectangular distortion (L1 = L2 ≠ L3). which preserves the total
volume than if the box remains undistorted (hint: if V is fixed and L1 = L2, then L3 =
V
L12 and L1 is the only "variable").
c. Show that the degree of distortion (ratio of L3 to L1) which will minimize the total
energy is L3 = 2 L1. How does this problem relate to Jahn-Teller distortions? Why
(in terms of the property of the central atom or ion) do we do the calculation with
fixed volume?
d. By how much (in eV) will distortion lower the energy (from its value for a cube, L1 =
L2 = L3) if V = 8 ?3 and h
2
8m = 6.01 x 10-27 erg cm2. 1 eV = 1.6 x 10-12 erg
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A particle on a ring model for electrons moving in cyclic compounds
12. The pi-orbitals of benzene, C6H6, may be modeled very crudely using the wave
functions and energies of a particle on a ring. Lets first treat the particle on a ring
problem and then extend it to the benzene system.
a. Suppose that a particle of mass m is constrained to move on a circle (of radius r) in
the xy plane. Further assume that the particle's potential energy is constant (choose
zero as this value). Write down the Schr?dinger equation in the normal Cartesian
coordinate representation. Transform this Schr?dinger equation to cylindrical
coordinates where x = rcosφ, y = rsinφ, and z = z (z = 0 in this case). Taking r to be
held constant, write down the general solution, Φ(φ), to this Schr?dinger equation.
The "boundary" conditions for this problem require that Φ(φ) = Φ(φ + 2pi). Apply
this boundary condition to the general solution. This results in the quantization of the
energy levels of this system. Write down the final expression for the normalized
wave functions and quantized energies. What is the physical significance of these
quantum numbers that can have both positive and negative values? Draw an energy
diagram representing the first five energy levels.
b. Treat the six pi-electrons of benzene as particles free to move on a ring of radius 1.40
?, and calculate the energy of the lowest electronic transition. Make sure the Pauli
15
principle is satisfied! What wavelength does this transition correspond to? Suggest
some reasons why this differs from the wavelength of the lowest observed transition
in benzene, which is 2600 ?.
A non-stationary state wave function
13. A diatomic molecule constrained to rotate on a flat surface can be modeled as a
planar rigid rotor (with eigenfunctions, Φ(φ), analogous to those of the particle on a ring
of problem 12) with fixed bond length r. At t = 0, the rotational (orientational)
probability distribution is observed to be described by a wave function Ψ(φ,0) = 43pi
Cos2φ. What values, and with what probabilities, of the rotational angular momentum,
??
?
??
?
-ih? ??φ , could be observed in this system? Explain whether these probabilities would be
time dependent as Ψ(φ,0) evolves into Ψ(φ,t).
A problem about Franck-Condon factors
14. Consider an N2 molecule, in the ground vibrational level of the ground electronic
state, which is bombarded by 100 eV electrons. This leads to ionization of the N2
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molecule to form N2+ . In this problem we will attempt to calculate the vibrational
distribution of the newly-formed N2+ ions, using a somewhat simplified approach.
a. Calculate (according to classical mechanics) the velocity (in cm/sec) of a 100 eV
electron, ignoring any relativistic effects. Also calculate the amount of time required
for a 100 eV electron to pass an N2 molecule, which you may estimate as having a
length of 2?.
b. The radial Schr?dinger equation for a diatomic molecule treating vibration as a
harmonic oscillator can be written as:
- h
?2
2μr2 ??
?
??
??
?r ??
?
??
?r2?Ψ
?r +
k
2(r - re) 2Ψ = E Ψ ,
Substituting Ψ(r) = F(r)r , this equation can be rewritten as:
-h
?2
2μ
?2
?r2 F(r) +
k
2(r - re) 2F(r) = E F(r) .
The vibrational Hamiltonian for the ground electronic state of the N2 molecule within this
approximation is given by:
H(N2) = -h
?2
2μ
d2
dr2 +
kN2
2 (r - rN2) 2 ,
where rN2 and kN2 have been measured experimentally to be:
rN2 = 1.09769 ?; kN2 = 2.294 x 106 gsec2 .
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The vibrational Hamiltonian for the N2+ ion , however, is given by :
H(N2) = -h
?2
2μ
d2
dr2 +
kN2+
2 (r - rN2+) 2 ,
where rN2+ and kN2+ have been measured experimentally to be:
rN2+ = 1.11642 ?; kN2+ = 2.009 x 106 gsec2 .
In both systems the reduced mass is μ = 1.1624 x 10-23 g. Use the above
information to write out the ground state vibrational wave functions of the N2 and
N2+ molecules, giving explicit values for any constants which appear in them.
The v = 0 harmonic oscillator function is Ψ0 = (α/pi)1/4 exp(-αx2/2).
c. During the time scale of the ionization event (which you calculated in part a),
the vibrational wave function of the N2 molecule has effectively no time to
change. As a result, the newly-formed N2+ ion finds itself in a vibrational state
which is not an eigenfunction of the new vibrational Hamiltonian, H(N2+ ).
Assuming that the N2 molecule was originally in its v=0 vibrational state,
calculate the probability that the N2+ ion will be produced in its v=0 vibrational
state.
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Vibration of a diatomic molecule
15. The force constant, k, of the C-O bond in carbon monoxide is 1.87 x 106 g/sec2.
Assume that the vibrational motion of CO is purely harmonic and use the reduced
mass μ = 6.857 amu.
Calculate the spacing between vibrational energy levels in this molecule,
in units of ergs and cm-1.
Calculate the uncertainty in the internuclear distance in this molecule,
assuming it is in its ground vibrational level. Use the ground state
vibrational wave function (Ψv=0; recall that I gave you this function in
problem 14), and calculate <x>, <x2>, and ?x = (<x2> - <x>2)1/2.
Under what circumstances (i.e. large or small values of k; large or small
values of μ) is the uncertainty in internuclear distance large? Can you
think of any relationship between this observation and the fact that helium
remains a liquid down to absolute zero?
A Variational Method Problem
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16. A particle of mass m moves in a one-dimensional potential whose Hamiltonian is
given by
H = - h
?2
2m
d2
dx2 + a|x| ,
where the absolute value function is defined by |x| = x if x ≥ 0 and |x| = -x if x ≤ 0.
a. Use the normalized trial wavefunction φ = ??
?
??
?2b
pi
1
4 e-bx2 to estimate the energy
of the ground state of this system, using the variational principle to evaluate W(b), the
variational expectation value of H.
b. Optimize b to obtain the best approximation to the ground state energy of this
system, using a trial function of the form of φ, as given above. The numerically
calculated exact ground state energy is 0.808616 h?
2
3 m-
1
3 a-
2
3 . What is the percent error
in your value?
Another Variational Method Problem
17. The harmonic oscillator is specified by the Hamiltonian:
H = - h
?2
2m
d2
dx2 +
1
2 kx2.
20
Suppose the ground state solution to this problem were unknown, and that you wish to
approximate it using the variational theorem. Choose as your trial wavefunction,
φ = 1516 a-
5
2 (a2 - x2) for -a < x < a
φ = 0 for |x| ≥ a
where a is an arbitrary parameter which specifies the range of the wavefunction. Note
that φ is properly normalized as given.
a. Calculate ??
-∞
+∞
φ*Hφdx and show it to be given by:
??
-∞
+∞
φ*Hφdx = 54 h
?2
ma2 +
ka2
14 .
b. Calculate ??
-∞
+∞
φ*Hφdx for a = b??
?
??
?h?2
km
1
4 .
c. To find the best approximation to the true wavefunction and its energy, find the
minimum of ??
-∞
+∞
φ*Hφdx by setting dda ??
-∞
+∞
φ*Hφdx = 0 and solving for a. Substitute this
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value into the expression for ??
-∞
+∞
φ*Hφdx given in part a. to obtain the best approximation
for the energy of the ground state of the harmonic oscillator.
d. What is the percent error in your calculated energy of part c.?
A Perturbation Theory Problem
18. Consider an electron constrained to move on the surface of a sphere of radius r0. The
Hamiltonian for such motion consists of a kinetic energy term only H0 = L
2
2mer02 , where
L is the orbital angular momentum operator involving derivatives with respect to the
spherical polar coordinates (θ,φ). H0 has the complete set of eigenfunctions ψ(0)lm =
Yl,m(θ,φ).
a. Compute the zeroth order energy levels of this system.
b. A uniform electric field is applied along the z-axis, introducing a perturbation
V = -eεz = -eεr0Cosθ , where ε is the strength of the field. Evaluate the correction to the
energy of the lowest level through second order in perturbation theory, using the identity
Cosθ Yl,m(θ,φ) = (l+m+1)(l-m+1)(2l+1)(2l+3) Yl+1,m(θ,φ) +
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(l+m)(l-m)
(2l+1)(2l-1) Yl-1,m(θ,φ) .
Note that this identity enables you to utilize the orthonormality of the spherical
harmonics.
c. The electric polarizability α gives the response of a molecule to an externally
applied electric field, and is defined by α = -?
2E
?2ε ??
?
ε=0
where E is the energy in the
presence of the field and ε is the strength of the field. Calculate α for this system.
d. Use this problem as a model to estimate the polarizability of a hydrogen atom,
where r0 = a0 = 0.529 ?, and a cesium atom, which has a single 6s electron with r0 ≈ 2.60
?. The corresponding experimental values are αH = 0.6668 ?3 and αCs = 59.6 ?3.
A Hartree-Fock problem you can do by hand
19. Given the following orbital energies (in hartrees) for the N atom and the coupling
elements between two like atoms (these coupling elements are the Fock matrix
elements from standard ab-initio minimum-basis SCF calculations), calculate the
molecular orbital energy levels and orbitals. Draw the orbital correlation diagram
for formation of the N2 molecule. Indicate the symmetry of each atomic and each
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molecular orbital. Designate each of the molecular orbitals as bonding, non-
bonding, or antibonding.
N1s = -15.31*
N2s = -0.86*
N2p = -0.48*
N2 σg Fock matrix*
??
??
??
??-6.52-6.22 -7.06
3.61 4.00 -3.92
N2 pig Fock matrix*
[ ]0.28
N2 σu Fock matrix*
??
??
??
??1.02-0.60 -7.59
0.02 7.42 -8.53
N2 piu Fock matrix*
[ ]-0.58
*The Fock matrices (and orbital energies) were generated using standard minimum basis
set SCF calculations. The Fock matrices are in the orthogonal basis formed from these
orbitals.
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An orbital correlation diagram problem
20. Given the following valence orbital energies for the C atom and H2 molecule, draw
the orbital correlation diagram for formation of the CH2 molecule (via a C2v insertion of
C into H2 resulting in bent CH2). Designate the symmetry of each atomic and molecular
orbital in both their highest point group symmetry and in that of the reaction path (C2v).
C1s = -10.91* H2 σg = -0.58*
C2s = -0.60* H2 σu = 0.67*
C2p = -0.33*
*The orbital energies were generated using standard STO3G minimum basis set SCF
calculations.
Practice using point group symmetry
21. Qualitatively analyze the electronic structure (orbital energies and orbitals) of
PF5. Analyze only the 3s and 3p electrons of P and the one 2p bonding electron of
each F. Proceed with a D3h analysis in the following manner:
a. Symmetry adapt the top and bottom F atomic orbitals.
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b. Symmetry adapt the three (trigonal) F atomic orbitals.
c. Symmetry adapt the P 3s and 3p atomic orbitals.
d. Allow these three sets of D3h orbitals to interact and draw the resultant
orbital energy diagram.
e. Symmetry label each of these molecular energy levels. Fill this energy
diagram with 10 "valence" electrons.
Practice with term symbols and determinental wave functions for atoms
and molecules
22. For the given orbital occupations (configurations) of the following systems,
determine all possible states (all possible allowed combinations of spin and space states).
There is no need to form the determinental wave functions, simply label each state with
its proper term symbol.
a.) CH21a122a121b223a111b11
26
b.) B2 1σg21σu22σg22σu21piu12piu1
c.) O2 1σg21σu22σg22σu21piu43σg21pig2
d.) Ti 1s22s22p63s23p64s23d14d1
e.) Ti 1s22s22p63s23p64s23d2
23. Construct Slater determinant wave functions for each of the following states of CH2:
a.) 1B1 (1a122a121b223a111b11)
b.) 3B1 (1a122a121b223a111b11)
c.) 1A1 (1a122a121b223a12)
A Woodward-Hoffmann rules problem
24. Let us investigate the reactions:
27
i. CH2(1A1) → H2 + C , and
ii. CH2(3B1) → H2 + C ,
under an assumed C2v reaction pathway utilizing the following information:
C atom: 3P ????????→
29.2 kcal/mole 1D
????????→
32.7 kcal/mole 1S
C(3P) + H2 → CH2(3B1) ?E = -78.8 kcal/mole
C(1D) + H2 → CH2(1A1) ?E = -97.0 kcal/mole
IP (H2) > IP (2s carbon).
a. Write down (first in terms of 2p1,0,-1 orbitals and then in terms of 2px,y,z
orbitals) the:
i. three Slater determinant (SD) wave functions belonging to the 3P state
all of which have MS = 1,
ii. five 1D SD wave functions, and
iii. one 1S SD wave function.
b. Using the coordinate system shown below, label the hydrogen orbitals σg, σu
and the carbon 2s, 2px, 2py, 2pz, orbitals as a1, b1(x), b2(y), or a2. Do the same
for the σ, σ, σ*, σ*, n, and ppi orbitals of CH2.
28
C
H
H
z
y
x
c. Draw an orbital correlation diagram for the CH2 → H2 + C reactions. Try to
represent the relative energy orderings of the orbitals correctly.
d. Draw a configuration correlation diagram for CH2(3B1) → H2 + C showing
all configurations which arise from the C(3P) + H2 products. You can assume
that doubly excited configurations lie much (~100 kcal/mole) above their parent
configurations.
e. Repeat step d. for CH2(1A1) → H2 + C again showing all configurations
which arise from the C(1D) + H2 products.
f. Do you expect the reaction C(3P) + H2 → CH2 to have a large activation
barrier? About how large? What state of CH2 is produced in this reaction?
29
Would distortions away from C2v symmetry be expected to raise or lower the
activation barrier? Show how one could estimate where along the reaction path
the barrier top occurs.
g. Would C(1D) + H2 → CH2 be expected to have a larger or smaller barrier
than you found for the 3P C reaction?
Another Woodward-Hoffmann rule problem
25. The decomposition of the ground-state singlet carbene,..
to produce acetylene and 1D carbon is known to occur with an activation energy
equal to the reaction endothermicity. However, when the corresponding triplet
carbene decomposes to acetylene and ground-state (triplet) carbon, the activation
energy exceeds this reaction's endothermicity. Construct orbital, configuration,
and state correlation diagrams that permit you to explain the above observations.
Indicate whether single configuration or configuration interaction wave functions
would be required to describe the above singlet and triplet decomposition
processes.
30
Practice with rotational spectrocopy and its relation to molecular
structure
26. Consider the molecules CCl4, CHCl3, and CH2Cl2.
What kind of rotor are they (symmetric top, etc; do not bother with oblate, or near-
prolate, etc.)
Will they show pure rotational (i.e., microwave) spectra?
27. Assume that ammonia shows a pure rotational spectrum. If the rotational constants
are 9.44 and 6.20 cm-1, use the energy expression:
E = (A - B) K2 + B J(J + 1),
to calculate the energies (in cm-1) of the first three lines (i.e., those with lowest K, J
quantum number for the absorbing level) in the absorption spectrum (ignoring higher
order terms in the energy expression).
A problem on vibration-rotation spectroscopy
31
28. The molecule 11B 16O has a vibrational frequency ωe = 1885 cm-1, a rotational
constant Be = 1.78 cm-1, and a bond energy from the bottom of the potential well of D0e =
8.28 eV. Use integral atomic masses in the following:
In the approximation that the molecule can be represented as a Morse oscillator,
calculate the bond length, Re in angstroms, the centrifugal distortion constant, De in
cm-1, the anharmonicity constant, ωexe in cm-1, the zero-point corrected bond energy,
D00 in eV, the vibration rotation interaction constant, αe in cm-1, and the vibrational
state specific rotation constants, B0 and B1 in cm-1. Use the vibration-rotation energy
expression for a Morse oscillator:
E = h_ ωe(v + 1/2) - h_ ωexe(v + 1/2)2 + BvJ(J + 1) - DeJ2(J + 1)2, where
Bv = Be - αe(v + 1/2), αe = -6Be
2
h_ωe +
6 Be3h_ωexe
h_ωe , and De =
4Be3
h_ωe2 .
Will this molecule show a pure rotation spectrum? A vibration-rotation
spectrum? Assuming that it does, what are the energies (in cm-1) of the first three
lines in the P branch (?v = +1, ?J = -1) of the fundamental absorption?
A problem labeling vibrational modes by symmetry
32
29. Consider trans-C2H2Cl2. The vibrational normal modes of this molecule are shown
below. What is the symmetry of the molecule? Label each of the modes with the
appropriate irreducible representation.
A problem in rotational spectroscopy
30. Suppose you are given two molecules (one is CH2 and the other is CH2- but you
33
don't know which is which). Both molecules have C2v symmetry. The CH bond
length of molecule I is 1.121 ? and for molecule II it is 1.076 ?. The bond angle of
molecule I is 104° and for molecule II it is 136°.
R
θ HH
y
z
C
a. Using a coordinate system centered on the C nucleus as shown above (the molecule is
in the YZ plane), compute the moment of inertia tensors of both species (I and II). The
definitions of the components of the tensor are, for example:
Ixx = ∑
j
mj(yj2 + zj2) - M(Y2 + Z2)
Ixy = -∑
j
mjxjyj - MXY
Here, mj is the mass of the nucleus j, M is the mass of the entire molecule, and X, Y, Z
are the coordinates of the center of mass of the molecule. Use ? for distances and amu's
for masses.
34
b. Find the principal moments of inertia Ia < Ib < Ic for both compounds ( in amu ?2
units) and convert these values into rotational constants A, B, and C in cm-1 using, for
example,
A = h(8pi2cIa)-1.
c. Both compounds are "nearly prolate tops" whose energy levels can be well
approximated using the prolate top formula:
E = (A - B) K2 + B J(J + 1),
if one uses for the B constant the average of the B and C valued determined earlier.
Thus, take the B and C values (for each compound) and average them to produce an
effective B constant to use in the above energy formula. Write down (in cm-1 units) the
energy formula for both species. What values are J and K allowed to assume? What is
the degeneracy of the level labeled by a given J and K?
d. Draw a picture of both compounds and show the directions of the three principle
axes (a,b,c). On these pictures, show the kind of rotational motion associated with the
quantum number K.
35
e. Suppose you are given the photoelectron spectrum of CH2-. In this spectrum Jj = Ji + 1
transitions are called R-branch absorptions and those obeying Jj = Ji - 1 are called P-
branch transitions. The spacing between lines can increase or decrease as functions of Ji
depending on the changes in the moment of inertia for the transition. If spacings grow
closer and closer, we say that the spectrum exhibits a so-called band head formation. In
the photoelectron spectrum that you are given, a rotational analysis of the vibrational
lines in this spectrum is carried out and it is found that the R-branches show band head
formation but the P-branches do not. Based on this information, determine which
compound I or II is the CH2- anion. Explain you reasoning.
f. At what J value (of the absorbing species) does the band head occur and at what
rotational energy difference?
Using point group symmetry in vibrational spectroscopy
31. Let us consider the vibrational motions of benzene. To consider all of the vibrational
modes of benzene we should attach a set of displacement vectors in the x, y, and z
directions to each atom in the molecule (giving 36 vectors in all), and evaluate how these
transform under the symmetry operations of D6h. For this problem, however, let's only
inquire about the C-H stretching vibrations.
36
a. Represent the C-H stretching motion on each C-H bond by an outward-directed vector
on each H atom, designated ri:
H
H
H
H
H
H
r2
r3
r4
r5
r6
r1
b. These vectors form the basis for a reducible representation. Evaluate the characters for
this reducible representation under the symmetry operations of the D6h group.
c. Decompose the reducible representation you obtained in part b. into its irreducible
components. These are the symmetries of the various C-H stretching vibrational modes
in benzene.
d. The vibrational state with zero quanta in each of the vibrational modes (the ground
vibrational state) of any molecule always belongs to the totally symmetric representation.
For benzene, the ground vibrational state is therefore of A1g symmetry. An excited state
which has one quantum of vibrational excitation in a mode which is of a given symmetry
37
species has the same symmetry species as the mode which is excited (because the
vibrational wave functions are given as Hermite polynomials in the stretching
coordinate). Thus, for example, excitation (by one quantum) of a vibrational mode of
A2u symmetry gives a wave function of A2u symmetry. To resolve the question of what
vibrational modes may be excited by the absorption of infrared radiation we must
examine the x, y, and z components of the transition dipole integral for initial and final
state wave functions ψi and ψf, respectively:
|< ψf | x | ψi >| , |< ψf | y | ψi >| , and |< ψf | z | ψi >| .
Using the information provided above, which of the C-H vibrational modes of benzene
will be infrared-active, and how will the transitions be polarized? How many C-H
vibrations will you observe in the infrared spectrum of benzene?
e. A vibrational mode will be active in Raman spectroscopy only if one or more of the
following integrals is nonzero:
|< ψf | xy | ψi >| , |< ψf | xz | ψi >| , |< ψf | yz | ψi >| ,
|< ψf | x2 | ψi >| , |< ψf | y2 | ψi >| , and |< ψf | z2 | ψi >| .
38
Using the fact that the quadratic operators transform according to the irreducible
representations:
(x2 + y2, z2) ? A1g
(xz, yz) ? E1g
(x2 - y2, xy) ? E2g
Determine which of the C-H vibrational modes will be Raman-active.
f. Are there any of the C-H stretching vibrational motions of benzene which cannot be
observed in either infrared of Raman spectroscopy? Give the irreducible representation
label for these unobservable modes.
A problem on electronic spectra and lifetimes
32. Time dependent perturbation theory provides an expression for the radiative lifetime
of an excited electronic state, given by τR:
τR = 3h
-4c3
4(Ei - Ef)3|μfi|2 ,
where i refers to the excited state, f refers to the lower state, and μfi is the transition
dipole.
39
a. Evaluate the z-component of the transition dipole for the 2pz → 1s transition in a
hydrogenic atom of nuclear charge Z, given:
ψ1s = 1pi ??? ???Za0
3
2 e
-Zr
a0 , and ψ2p
z =
1
4 2pi ?
?? ???Za
0
5
2 r Cosθ e
-Zr
2a0 .
Express your answer in units of ea0.
b. Use symmetry to demonstrate that the x- and y-components of μfi are zero, i.e.
<2pz| e x |1s> = <2pz| e y |1s> = 0.
c. Calculate the radiative lifetime τR of a hydrogenlike atom in its 2pz state. Use the
relation e2 = h
-2
mea0 to simplify your results.
The difference between slowly and quickly turning on a perturbation
33. Consider a case in which the complete set of states {φk} for a Hamiltonian is known.
a. If the system is initially in the state m at time t=0 when a constant perturbation V is
suddenly turned on, find the probability amplitudes Ck(2)(t) and Cm(2)(t), to second order
in V, that describe the system being in a different state k or the same state m at time t.
40
b. If the perturbation is turned on adiabatically (i.e., very slowly), what are Ck(2)(t) and
Cm(2)(t)? Here, consider that the initial time is t0 → -∞, and the potential is V eηt, where
the positive parameter η is allowed to approach zero η→ 0 in order to describe the
adiabatically turned on perturbation.
c. Compare the results of parts a. and b. and explain any differences.
d. Ignore first order contributions (assume they vanish) and evaluate the transition rates
ddt |Ck(2)(t)|2 for the results of part b. by taking the limit η → 0+, to obtain the adiabatic
results.
An example of quickly turning on a perturbation- the sudden
approximation
34. Consider an interaction or perturbation which is carried out suddenly
(instantaneously, e.g., within an interval of time ?t which is small compared to the
natural period ωnm-1 corresponding to the transition from state m to state n), and after
that is turned off adiabatically (i.e., extremely slowly as V eηt). The transition
probability in this case is given as:
41
Tnm ≈ |<n|V|m>|
2
h-2ωnm2
where V corresponds to the maximum value of the interaction when it is turned on. This
formula allows one to calculate the transition probabilities under the action of sudden
perturbations which are small in absolute value whenever perturbation theory is
applicable.
Let's use this "sudden approximation" to calculate the probability of excitation of
an electron under a sudden change of the charge of the nucleus. Consider the reaction:
1
3 H →
2
3 He+ + e-,
and assume the tritium atom has its electron initially in a 1s orbital.
a. Calculate the transition probability for the transition 1s → 2s for this reaction using the
above formula for the transition probability.
b. Suppose that at time t = 0 the system is in a state which corresponds to the
wave function ?m, which is an eigenfunction of the operator H0. At t = 0, the sudden
change of the Hamiltonian occurs (now denoted as H and remains unchanged). Calculate
the same 1s → 2s transition probability as in part a., only this time as the square of the
magnitude of the coefficient, A1s,2s using the expansion:
42
Ψ(r,0) = ?m(r) = ∑
n
Anmψn(r) , where Anm = ???m(r)ψn(r)d3r
Note, that the eigenfunctions of H are ψn with eigenvalues En. Compare this value with
that obtained by perturbation theory in part a.
A symmetric top rotational spectrum problem
35. The methyl iodide molecule is studied using microwave (pure rotational)
spectroscopy. The following integral governs the rotational selection rules for transitions
labeled J, M, K → J', M', K':
I = <DM'K'J' | ε→ . μ→ |DMKJ >.
The dipole moment μ→ lies along the molecule's C3 symmetry axis. Let the electric field
of the light ε→ define the lab-fixed Z-direction.
a. Using the fact that Cosβ = D001* , show that
I = 8pi2με(-1)(M+K)?? ??M 0 MJ' 1 J ?? ??K 0 KJ' 1 J δM'MδK'K
b. What restrictions does this result place on ?J = J' - J? Explain physically why the K
quantum number can not change.
43
A problem in electronic and photo-electron spectroscopy
36. Consider the molecule BO.
a. What are the total number of possible electronic states that can be formed by
combination of ground-state B and O atoms?
b. What electron configurations of the molecule are likely to be low in energy?
Consider all reasonable orderings of the molecular orbitals. What are the states
corresponding to these configurations?
c. What are the bond orders in each of these states?
d. The true ground state of BO is 2Σ. Specify the +/- and u/g symmetries for this
state.
e. Which of the excited states you derived above will radiate to the 2Σ ground state?
Consider electric dipole radiation only.
f. Does ionization of the molecule to form a cation lead to a stronger, weaker, or
equivalent bond strength?
44
g. Assuming that the energies of the molecular orbitals do not change upon
ionization, what are the ground state, the first excited state, and the second excited
state of the positive ion?
h. Considering only these states, predict the structure of the photoelectron spectrum
you would obtain for ionization of BO.
A problem on vibration-rotation spectroscopy
37.
-1
600 cm 800 cm 1300 cm 1500 cm 3200 cm 3600 cm
-1 -1 -1
-1
-1
2ν (HCN)
2
ν (HCN)
3
3317 cm
-1
ν (HCN)
2
712 cm
-1
45
The above figure shows part of the infrared absorption spectrum of HCN gas. The
molecule has a CH stretching vibration, a bending vibration, and a CN stretching
vibration.
a. Are any of the vibrations of linear HCN degenerate?
b. To which vibration does the group of peaks between 600 cm-1 and 800 cm-1 belong?
c. To which vibration does the group of peaks between 3200 cm-1 and 3400 cm-1 belong?
d. What are the symmetries (σ, pi, δ) of the CH stretch, CN stretch, and bending
vibrational motions?
e. Starting with HCN in its 0,0,0 vibrational level, which fundamental transitions would
be infrared active under parallel polarized light (i.e., z-axis polarization):
000 → 001?
000 → 100?
000 →010?
f. Why does the 712 cm-1 transition have a Q-branch, whereas that near 3317 cm-1 has
only P- and R-branches?
46
A Problem in Which You Can Practice Deriving Equations
This is Important Because a Theory Scientist Does Derivations as Part
of Her/His Job
38.
By expanding the molecular orbitals {φk} as linear combinations of atomic orbitals {χμ},
φk = ∑
μ
cμkχμ
show how the canonical Hartree-Fock (HF) equations:
F φi = εi φj
reduce to the matrix eigenvalue-type equation of the form:
∑
ν
FμνCνi = εi∑
ν
SμνCνi
where:
Fμν = < >χμ| |h χν + ∑
δ κ
??? ???γδκ< >χμχδ| |g χνχκ - γδκex< >χμχδ| |g χκχν ,
Sμν = < >χμ|χν , γδκ = ∑
i=occ
CδiCκi ,
and
47
γδκex = ∑
i= occ andsame spin
CδiCκi .
Note that the sum over i in γδκ and γδκex is a sum over spin orbitals. In addition, show
that this Fock matrix can be further reduced for the closed shell case to:
Fμν = < >χμ| |h χν + ∑
δ κ
Pδκ??? ???< >χμχδ| |g χνχκ - 12 < >χμχδ| |g χκχν ,
where the charge bond order matrix, P, is defined to be:
Pδκ = ∑
i=occ
2CδiCκi ,
where the sum over i here is a sum over orbitals not spin orbitals.
Another Derivation Practice Problem
39. Show that the HF total energy for a closed-shell system may be written in terms of
integrals over the orthonormal HF orbitals as:
E(SCF) = 2 ∑
k
occ
< >φk| |h φk + ∑
kl
occ
??? ???2< >kl| |g kl - < >kl| |g lk +
∑
μ>ν
ZμZνRμν .
48
One More Derivation Problem
40. Show that the HF total energy may alternatively be expressed as:
E(SCF) = ∑
k
occ
??? ???εk + < >φk| |h φk + ∑
μ>ν
ZμZνRμν ,
where the εk refer to the HF orbital energies.
A Molecular Hartree-Fock SCF Problem
41. This problem will be concerned with carrying out an SCF calculation for the HeH+
molecule in the 1Σg+(1σ2) ground state. The one- and two-electron integrals (in atomic
units) needed to carry out this SCF calculation at R = 1.4 a.u. using Slater type orbitals
with orbital exponents of 1.6875 and 1.0 for the He and H, respectively are:
S11 = 1.0, S22 = 1.0, S12 = 0.5784,
h11 = -2.6442, h22 = -1.7201, h12 = -1.5113,
g1111 = 1.0547, g1121 = 0.4744, g1212 = 0.5664,
g2211 = 0.2469, g2221 = 0.3504, g2222 = 0.6250,
49
where 1 refers to 1sHe and 2 to 1sH. The two-electron integrals are given in Dirac
notation. Parts a. – d should be done by hand. Any subsequent parts can make use of the
QMIC software that can be found at
http://www.emsl.pnl.gov:2080/people/bionames/ja_nichols.html.
a. Using φ1 ≈ 1sHe for the initial guess of the occupied molecular orbital, form a
2x2 Fock matrix. Use the equation derived above in problem 38 for Fμν.
b. Solve the Fock matrix eigenvalue equations given above to obtain the orbital
energies and an improved occupied molecular orbital. In so doing, note that < >φ1|φ1 =
1 = C1TSC1 gives the needed normalization condition for the expansion coefficients of
the φ1 in the atomic orbital basis.
c. Determine the total SCF energy using the expression of problem 39 at this step
of the iterative procedure. When will this energy agree with that obtained by using the
alternative expression for E(SCF) given in problem 40?
d. Obtain the new molecular orbital, φ1, from the solution of the matrix
eigenvalue problem (part b).
e. A new Fock matrix and related total energy can be obtained with this improved
choice of molecular orbital, φ1. This process can be continued until a convergence
criterion has been satisfied. Typical convergence criteria include: no significant change
in the molecular orbitals or the total energy (or both) from one iteration to the next.
Perform this iterative procedure for the HeH+ system until the difference in total energy
between two successive iterations is less than 10-5 a.u.
50
f. Show, by comparing the difference between the SCF total energy at one
iteration and the converged SCF total energy, that the convergence of the above SCF
approach is primarily linear (or first order).
g. Is the SCF total energy calculated at each iteration of the above SCF procedure
as in problem 39 an upper bound to the exact ground-state total energy?
h. Does this SCF wave function give rise (at R→∞) to proper dissociation
products?
A Configuration Interaction Problem
42. This problem will continue to address the same HeH+ molecular system as above,
extending the analysis to include correlation effects. We will use the one- and two-
electron integrals (same geometry) in the converged (to 10-5 au) SCF molecular orbital
basis which we would have obtained after 7 iterations above. The converged mos you
should have obtained in problem 1 are:
φ1 = ???
?
??
??-0.89997792
-0.15843012 φ2 = ??
??
??
??-0.83233180
1.21558030
a. Carry out a two configuration CI calculation using the 1σ2 and 2σ2
configurations first by obtaining an expression for the CI matrix elements HI,J (I,J = 1σ2,
2σ2) in terms of one- and two-electron integrals, and secondly by showing that the
resultant CI matrix is (ignoring the nuclear repulsion energy):
51
??
??
??
??-4.2720 0.1261
0.1261 -2.0149
b. Obtain the two CI energies and eigenvectors for the matrix found in part a.
c. Show that the lowest energy CI wave function is equivalent to the following
two-determinant (single configuration) wave function:
1
2 ?
?? ?????? ?????? ???a12φ
1 + b
1
2φ
2 α?
?? ???a12φ
1 - b
1
2φ
2 β + ?
?? ?????? ???a12φ
1 - b
1
2φ
2 α?
?? ???a12φ
1 + b
1
2φ
2 β
involving the polarized orbitals: a
1
2 φ
1 ± b
1
2 φ
2 , where a = 0.9984 and b = 0.0556.
d. Expand the CI list to 3 configurations by adding1σ2σ to the original 1σ2 and
2σ2 configurations of part a above. First, express the proper singlet spin-coupled 1σ2σ
configuration as a combination of Slater determinants and then compute all elements of
this 3x3 matrix.
e. Obtain all eigenenergies and corresponding normalized eigenvectors for this CI
problem.
f. Determine the excitation energies and transition moments for HeH+ using the
full CI result of part e above. The nonvanishing matrix elements of the dipole operator
r(x,y,z) in the atomic basis are:
< >1sH| |z 1sHe = 0.2854 and < >1sH| |z 1sH = 1.4.
First determine the matrix elements of r in the SCF orbital basis then determine the
excitation energies and transition moments from the ground state to the two excited
singlet states of HeH+.
52
g. Now turning to perturbation theory, carry out a perturbation theory calculation
of the first-order wave function |1σ2>(1) for the case in which the zeroth-order wave
function is taken to be the 1σ2 Slater determinant. Show that the first-order wave
function is given by:
|1σ2>(1) = -0.0442|2σ2>.
h. Why does the |1σ2σ> configuration not enter into the first-order wave
function?
i. Normalize the resultant wave function that contains zeroth- plus first-order parts
and compare it to the wave function obtained in the two-configuration CI study of part b.
j. Show that the second-order RSPT correlation energy, E(2), of HeH+ is -0.0056
a.u. How does this compare with the correlation energy obtained from the two-
configuration CI study of part b?
Repeating the SCF Problem but With a Computer Program
43. Using either programs that are available to you or the QMIC programs that you can
find at the web site
http://www.emsl.pnl.gov:2080/people/bionames/nichols_ja.html
calculate the SCF energy of HeH+ using the same geometry as in problem 42 and the
STO3G basis set provided in the QMIC basis set library. How does this energy compare
53
to that found in problem 42? Run the calculation again with the 3-21G basis basis
provided. How does this energy compare to the STO3G and the energy found using
STOs in problem 42?
A Series of SCF Calculations to Produce a Potential Energy Curve
44. Generate SCF potential energy surfaces for HeH+ and H2 using the QMIC software
or your own programs. Use the 3-21G basis set and generate points for geometries of R =
1.0, 1.2, 1.4, 1.6, 1.8, 2.0, 2.5, and 10.0 a0. Plot the energies vs. geometry for each
system. Which system dissociates properly?
Configuration Interaction Potential Curves for Several States
45. Generate CI potential energy surfaces for the 4 states of H2 resulting from a
calculation with 2 electrons occupying the lowest 2 SCF orbitals (1σg and 1σu) in all
possible ways. Use the same geometries and basis set as in problem 44. Plot the energies
vs. geometry for each system. Properly label and characterize each of the states (e.g.,
repulsive, dissociate properly, etc.).
A Problem on Partition Functions and Thermodynamic Properties
54
46. F atoms have 1s22s22p5 2P ground electronic states that are split by spin-orbit
coupling into 2P3/2 and 2P1/2 states that differ by only 0.05 eV in energy.
a. Write the electronic partition function (take the energy of the 2P3/2 state to be zero and
that of the 2P1/2 state to be 0.05eV and ignore all other states) for each F atom.
b. Using
E = kT
2
? lnQ
?T
?
?
?
?
?
N ,V
, derive an expression for the average electronic energy
E of N gaseous F atoms.
c. Using the fact that kT=0.03eV at T=300°K, make a (qualitative) graph of E /N vs T for
T ranging from 100°K to 3000°K.
A Problem Using Transition State Theory
47. Suppose that we used transition state theory to study the reaction
NO(g) + Cl2(g) → NOCl(g) + Cl(g) assuming it to proceed through a bent transition
state, and we obtained an expression for the rate coefficient
k
bent
=
kT
h
e
?E
≠
/kT
q
≠
/ v( )
q
NO( )
v
q
Cl2
( )
v
55
a. Now, let us consider what differences would occur if the transition state structure were
linear rather than bent. Assuming that the activation energy E≠ and electronic state
degeneracies are not altered, derive an expression for the ratio of the rate coefficients for
the linear and bent transition state cases
k
linear
k
bent
=
b. Using the following order of magnitude estimates of translational, rotational, and
vibrational partition functions per degree of freedom at 300°K
qt ~ 108, qr ~ 102, qv ~ 1,
what ratio would you expect for klinear/kbent?
A Problem With Slater Determinants
48. Show that the configuration (determinant) corresponding to the Li+ 1s(α)1s(α) state
vanishes.
56
Another Problem With Slater Determinants and Angular Momenta
49. Construct the 3 triplet and 1 singlet wave functions for the Li+ 1s12s1 configuration.
Show that each state is a proper eigenfunction of S2 and Sz (use raising and lowering
operators for S2)
A Problem With Slater Determinants for a Linear Molecule
50. Construct determinant wave functions for each state of the 1σ22σ23σ21pi2
configuration of NH.
A Problem With Slater Determinants for an Atom
51. Construct determinant wave functions for each state of the 1s12s13s1 configuration
of Li.
A Problem on Angular Momentum of an Atom
52. Determine all term symbols that arise from the 1s22s22p23d1 configuration of the
excited N atom.
Practice With the Slater Condon Rules
53. Calculate the energy (using Slater Condon rules) associated with the 2p valence
electrons for the following states of the C atom.
i. 3P(ML=1,MS=1),
57
ii. 3P(ML=0,MS=0),
iii. 1S(ML=0,MS=0), and
iv. 1D(ML=0,MS=0).
More Practice With the Slater Condon Rules
54. Calculate the energy (using Slater Condon rules) associated with the pi valence
electrons for the following states of the NH molecule.
i. 1? (ML=2, MS=0),
ii. 1Σ (ML=0, MS=0), and
iii. 3Σ (ML=0, MS=0).
Practice With The Equations of Statistical Mechanics
55. Match each of the equations below with the proper phrase A-K
58
B
2
= ?2pi r
2
o
∞
∫ e
?u(r)/ kT
? 1( )dr
E
2
? E ( )
2
= kT
2
?E
?T
?
?
?
?
?
N ,V
2pimkT
h
2
?
?
?
?
Q = e
?
Nφ
2kT
e
?θ /2T
1 ? e
? θ /T
?
?
?
?
?
?
3N
g ν( ) = αν
2
Q =
M !
N !(M ? N)!
q
N
Θ =
qe
μo/kT
p
1 + qe
μo /kT
p
p
A
= p
A
o
X
A
59
cω
kT
= ?4
W=WAANAA+WBBNBB+WABNAB
N
AB
?
N
A
cN
B
N
A
+N
B
A. Raoult's law
B. Debye solid
C. Critical Point
D. Ideal adsorption
E. Langmuir isotherm
F. Bragg-Williams
G. Partition function for surface translation
H. Concentrated solution
I. Fluctuation
J. Virial coefficient
60
K. Einstein solid
A Problem Dealing With the Second Virial Coefficient
56. The Van der Waals equation of state is
p +
N
V
?
?
?
?
?
2
a
?
?
?
?
?
? V ? Nb( ) = NkT
solve this equation for p, and then obtain an expression for
pV
NkT
. Finally, expand
pV
NkT
in powers of
N
V
?
?
?
?
?
and obtain an expression for the second virial coefficient of this Van
der Waals gas in terms of b, a, and T.
A Problem to Make You Think About Carrying Out Monte-Carlo and
Molecular Dynamics Simulations
57. Briefly answer each of the following:
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For which of the following would you be wisest to use Monte-Carlo (MC) simulation and
for which should you use molecular dynamics (MD)
a. Determining the rate of diffusion of CH4 in liquid Kr.
b. Determining the equilibrium radial distribution of Kr atoms relative to the CH4 in the
above example
c. Determining the mean square end-to-end distance for a floppy hydrocarbon chain in
the liquid state
Suppose you are carrying out a Monte-Carlo simulation involving 1000 Ar atoms.
Further suppose that the potentials are pair wise additive and that your computer requires
approximately 50 floating point operations (FPO's) (e.g. multiply, add, divide, etc.) to
compute the interaction potential between any pair of atoms
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d. For each M-C trial move, how many FPO's are required? Assuming your computer
has a speed of 100 MFlops (i.e., 100 million FPO's per sec), how long will it take you to
carry out 1,000,000 M-C moves?
e. If the fluctuations observed in the calculation of question d are too large, and you wish
to make a longer M-C calculation to reduce the statistical "noise", how long will your
new calculation require if you wish to cut the noise in half?
f. How long would the calculation of question d require if you were to use 1,000,000 Ar
atoms (with the same potential and the same computer)?
g. Assuming that the evaluation of the forces between pairs of Ar atoms (?V/?r;) requires
approximately the same number of FPO's (50) as for computing the pair potential, how
long (in sec) would it take to carry out a molecular dynamics simulation involving 1000
Ar atoms using a time step (Dt) of 10-15 sec and persisting for a total time duration of
one nanosecond (10-9 sec) using the 100 MFlop computer?
h. How long would a 10-6 MFlop (i.e., 1 FPO per sec) Ph.D. student take to do the
calculation in part d?
A Problem to Practice Using Partition Functions
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58. In this problem, you will compute the pressure-unit equilibrium constant Kp for the
equilibrium
2 Na ? Na2
in the gas phase at a temperature of 1000 K. Your final answer should be expressed in
units of atm-1. In doing so, you need to consider the electronic term symbols of Na and of
Na2, and you will need to use the following data:
i. Na has no excited electronic states that you need to consider.
ii. (h2/8pi2Ik) = 0.221 K for Na2
iii. (hν/k) = 229 K for Na2
iv. 1 atm = 1.01 x106 dynes cm-2
v. The dissociation energy of Na2 from the v = 0 to dissociation is D0 = 17.3 kcal mol-1.
a. First, write the expressions for the Na and Na2 partition functions showing their
translational, rotational, vibrational and electronic contributions.
b. Next, substitute the data and compute Kp, and change units to atm-1.
A Problem Using Transition State Theory
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59. Looking back at the NO + Cl2 reaction treated using transition state theory in
Problem 47, let us assume that this same reaction (via. the bent transition state) were to
occur while to reagents NO and Cl2 were adsorbed to a surface in the following manner:
a. both NO and Cl2 lie flat against the surface with both of their atoms touching the
surface.
b. both NO and Cl2 move freely along the surface (i.e., they can translate parallel to the
surface).
c. both NO and Cl2 are tightly bound to the surface in a manner that causes their
movements perpendicular to the surface to become high-frequency vibrations.
Given this information, and again assuming the following order of magnitude estimates
of
partition functions
qt ~ 108, qr ~ 102, qv ~ 1
calculate the ratio of the TS rate constants for this reaction occurring in the surface
adsorbed state and in the gas phase. In doing so, you may assume that the activation
energy and all properties of the transition state are identical in the gas and adsorbed state,
except that the TS species is constrained to lie flat on the surface just as are NO and Cl2
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